Accessibility properties of abnormal geodesics in optimal control illustrated by two case studies

Figure 1.  Unfolding semicubical cusp

Figure 2.  Quickest nautical path as a miniversal unfolding of the generic singularity of the abnormal geodesic

Figure 3.  (top) Singular vector field $ X_s $ on the surface $ D''(\tilde q) = 0 $ for the semi-normal form constructed in the proof of Proposition 2.8 for $ \delta_1 = \delta_2 = 0 $ and random values for the others coefficients $ a_{ij}, b_{ij}, c_{ij} $ yielding $ \lambda = -4 $. (bottom) Corresponding strong (blue) and weak (white) current domains. The curve $ \{\|F_0(q)\|_g = 1\} $ is not regular

Figure 4.  Hyperbolic and abnormal geodesics in a neighbourhood of the collinearity set

Figure 5.  Local synthesis near $ \mathcal{E} $ in the generic case

Figure 6.  Local synthesis near $ \mathcal{E} $ in the codimension $ 2 $ case for $ b<0 $. The dashed curves are in the half-space $ x<0 $

Figure 7.  Local synthesis near $ \mathcal{E} $ in the codimension 2 case for $ b>0 $

Figure 8.  Strata of the surface $ S $ separating the regions of $ U $ where the control is $ \pm 1 $ for the model (18) with $ b = b_1 = 1, \ c = 0 $. We also represent the regions where $ \sigma_\pm $ intersect $ N $ for $ t<0 $ and several trajectories which generate a switching locus and a splitting locus

Figure 9.  Minimum time $ t^* = \max(t_1^{\varepsilon}, t_2^{\varepsilon}, t_3) $ to reach $ (0, w, s)\in N $ from a neighbourhood $ U $ of $ 0 $ for the model (18) with $ b = b_1 = 1, \ = c = 0 $. The exceptional locus is $ \mathcal{E}: y = -z^2 $ and the singular locus is $ \mathcal{S}: n\cdot [Y, X](q) = 0: z = 0 $